1. Field of the Invention
The field of art to which this invention relates is motion analysis, and more particularly to geometric methods for analyzing shape changes resulting from repetitive motions using Elliptic Fourier Analysis (EFA).
2. Description of the Related Art
Geometric morphometric methods characterize the shape of a configuration of landmarks in such a way as to retain all of the relative spatial information encoded in the original data throughout an analysis (See, Slice, D. E., F. L. Bookstein, L. F. Marcus, and F. J. Rohlf. 1996. A glossary for geometric morphometrics. In L. F. Marcus, M. Corti, A. Loy, G. J. P. Naylor and D. E. Slice (eds.), Advances in Morphometrics, pp. 531-551. Plenum, N.Y.). Often this information is represented as a matrix of the coordinates of points, called landmarks, that are presumed to capture the shape of the structure and to be homologous across specimens within a study. Such raw data cannot be analyzed directly since the values of the coordinates are a function of the location and orientation of the specimen, with respect to the digitizing device, at the time the data were collected. Geometric morphometric analyses register all specimens to a common coordinate system to remove such effects. This variation is usually further decomposed into components of size and shape differences.
Specifically, let Xi be a pxc3x97k matrix of the k coordinates of the p landmarks describing the shape of the ith specimen. The prior art geometric morphometric analysis of such data usually begins by fitting the model:
Xxe2x80x2i=xcex1i(XC+Di)Hi+1xcfx84i
where xcex1i is a scalar scale factor, Hi is an orthonormal kxc3x97k rotation matrix, 1 is a pxc3x971 vector of 1s, xcfx84i is a 1xc3x97k vector of translation terms, and Di is a pxc3x97k matrix of deviations from a pxc3x97k consensus configuration XC. The parameters of the model are estimated for each specimen so as to minimize the trace of (Xxe2x80x2ixe2x88x92XC)t(Xxe2x80x2ixe2x88x92XC) subject to the constraints that Xxe2x80x2it1=0, where 0 is a kxc3x971 vector of 0s, and tr(Xxe2x80x2itXxe2x80x2i)=1. The constraints simply mean that each Xxe2x80x2i is centered at the origin and that the sum of squared distances of the landmarks from the origin equals unity. The criterion being minimized is the sum of squared Euclidean distances from each landmark on Xxe2x80x2i to the corresponding landmark on XC. Since XC is unknown, it and the parameters of the model are estimated by an iterative process using one of the Xi as an initial estimate of XC (See, Gower, J. C. 1975. Generalized Procrustes analysis. Psychometrika, 40:33-51; and Rohlf, F. J., and D. E. Slice. 1990. Extensions of the Procrustes method for the optimal superimposition of landmarks. Systematic Zoology, 39:40-59).
The original data required pk parameters to represent variation in each of the k coordinates at each of the p points in a configuration. The fitting and associated constraints, however, impose a certain structure on the data that cannot be ignored during subsequent analyses. One degree of freedom in the sample variation is lost due to the estimation of the scale parameter, k degrees of freedom are lost due to translation, and k(kxe2x88x921)/2 due to rotation. Though the superimposed data are still represented by pk values, their variation has, at most, pk-k-k(kxe2x88x921)/2xe2x88x921 degrees of freedom. Furthermore, this reduced shape space is non-Euclidean (See, Kendall, D. G. 1984. Shape manifolds, Procrustean metrics, and complex projective spaces. Bull. Lond. Math. Soc., 16:81-121; and Kendall, D. G. 1985. Exact distributions for shapes of random triangles in convex sets. Adv. Appl. Probab., 17:308-329), i.e. it is curved, thus precluding the direct application of standard linear statistical analyses. To address the latter problem, data are usually projected into a linear space tangent to shape space at the point of XC. Such an operation provides the best linear approximation to the curved shape space (See, Rohlf, F. J. 1996. Morphometric spaces, shape components, and the effects of linear transformations. In L. F. Marcus, M. Corti, A. Loy, G. J. P. Naylor and D. E. Slice (eds.), Advances in Morphometrics, pp. 117-129. Plenum, N.Y.).
A collection of triangles in two dimensions provides a simple, low-dimensional example. Each triangle can be completely described by the x,y coordinates of its three vertices and represented as a point in a pxc3x97k=2xc3x973=6 dimensional space of all triangles. Superimposing the sample on their mean configuration using the above procedure results in a loss of k-k(kxe2x88x921)/2xe2x88x921=4 degrees of freedom, leaving two degrees of freedom for shape variation. For triangles, this curving 2D space embedded within the original six dimensional space can be visualized as the surface of a hemisphere centered at the origin. For small amounts of variation, the projection of the scatter onto a tangent plane touching the surface of this hemisphere at the point representing XC can be used as a linear approximation of the variation in shape space. Such an approximation of a curving surface by a planar one is analogous to using points on a flat map to represent positions on the curving surface of the earth. For configurations of more 2D points, shape space is a complex (pxe2x88x922)-torus. The situation for configurations of higher dimension is comparable, but the structure of the shape space and the associated mathematics are much more complicated (See, Goodall, C. R. 1992. Shape and image analysis for industry and medicine. Short course. University of Leeds, Leeds, UK).
Geometric morphometrics provides a sophisticated suite of methods for the processing and analysis of shape data (see, Bookstein, F. L. 1991. Morphometric Tools for Landmark Data. Geometry and Biology. Cambridge University Press, New York; Rohlf, F. J., and L. F. Marcus. 1993. A revolution in morphometrics. Trends in Ecology and Evolution, 8:129-132; and Marcus, L. F., and M. Corti. 1996. Overview of the new, or geometric morphometrics. In L. F. Marcus, M. Corti, A. Loy, G. J. P. Naylor and D. E. Slice (eds.), Advances in Morphometrics, pp. 1-13. Plenum, N. Y.). Most geometric morphometric analyses to date have been oriented toward assessing group differences and the covariation of shape with extrinsic variables (see Marcus, L. F., M. Corti, A. Loy, G. J. P. Naylor, and D. E. Slice. 1996. Advances in Morphometrics, NATO ASI Series A: Life Sciences, pp. xiv+587. Plenum Press, N.Y. for numerous examples). For many questions, though, such xe2x80x9cstaticxe2x80x9d analyses are inadequate. The study of feeding, flying, walking, swinging, or swimming, for instance, require methods capable of characterizing dynamic, repetitive changes in the shape of a single set of structures within each specimen.
For practical applications, complex motions involving many landmarks can be analyzed. However, a simple data set can be used to illustrate the inadequacy of standard methods and the efficacy of the new procedure for the analysis of motion-related shape change.
FIGS. 1A-1C show triangular configurations associated with three individuals of three hypothetical xe2x80x9cspeciesxe2x80x9d used to model the changes in the relative locations of points associated with a particular motion. It is hypothesized that during the course of the motion, point C moves with respect to points A and B. Which points move is irrelevant in shape analysis, and, in fact, cannot be determined from the shape differences alone.
In the species brevistrokus, shown in FIG. 1A, the xe2x80x9cidealizedxe2x80x9d motion has point C moving a short distance back and forth at a right angle to the line segment connecting points A and B. In the second species, longistrokus, shown in FIG. 1B, point C moves in approximately the same direction, but the range of motion is twice that of brevistrokus. In the last species, elliptistrokus, shown in FIG. 1C, the motion of point C is in the same general direction as in the other two species, but follows an elliptical path. The individual in the sample expressing the idealized motions for a species is referred to as the xe2x80x9cmodelxe2x80x9d specimen.
Slight deviations in the idealized motions just described were drawn by hand to simulate individual variation within a species (dotted lines in FIGS. 1A-1C). For the two species expressing linear motion in point C (brevistrokus and longistrokus), one individual was constructed with a slightly curving deviation from the idealized motion and another with a slightly angular deviation. For elliptistrokus, one individual was constructed with a longer, narrower elliptical stroke than ideal and one with a shorter, fatter stroke.
To generate data for analysis, a drawing of each individual is digitized three times at ten points within one realization of the hypothetical motion. For each set of ten digitized configurations, care is taken to ensure that all ten configurations are in sequence with respect to the cycle of motion, however, the starting point within the motion is varied slightly. Otherwise, digitizing is done with intentional imprecision to introduce additional variation due to measurement error. This process results in a data set of 270 individual configurations representing ten configurations taken from three replicate motions of three individuals from each of the three hypothetical species.
The data is first superimposed by generalized least-squares superimposition (GLS) method using Morpheus et al. (See, Slice, D. E. 1998. Morpheus et al.: Software for Morphometric Research. Revision 01-30-98. Department of Ecology and Evolution, State University of New York at Stony Brook, Stony Brook, N.Y.). As described previously, this mapped the six-dimensional data into a two-dimensional subspace. That this space is curved, rather than a plane, is evidenced by the results of a singular value decomposition (a short cut to a principal components analysis) of the mean-centered superimposed data matrix carried out using NTSYSpc (See, Rohlf, F. J. 1997. NTSYS-pc: Numerical Taxonomy and Multivariate Analysis System. Exeter Software, Setauket, N.Y.). Three of the principal component (PC) axes were associated with nonzero variation in the superimposed sample instead of two as would be expected for planar variation. Projections of the data onto these axes are shown in FIGS. 2A and 2B, where FIG. 2A shows the data plotted on axes associated with the largest and smallest nonzero sample variation. FIG. 2B shows the data plotted with respect to axes associated with the largest and second largest amount variation. FIG. 2A is an edge-on view of the subspace revealing its curvature. FIG. 2B is a tangent space approximation of the scatter in the curved subspace.
For this data, the application of standard exploratory methods fail to provide the desired results. A UPGMA clustering of all 270 configurations, for instance, showed no structure indicative of the true relationships between the configurations. Neither did the UPGMA clustering of mean shapes of individual replicate motions reflect the known structure within the sample. Even replicates of some individuals in the latter analysis were indicated as being quite distinct, with no discernable pattern suggesting a possible explanation.
These results illustrate the need for the incorporation of known information about the relationships of configurations to the motion into the analysis of this type of data.
To overcome the disadvantages of the motion analysis methods of the prior art, the present invention provides a geometric method for the analysis of shape changes resulting from repetitive motions, such as those that arise in studies of swimming, walking, jumping, or feeding. Specifically, a multivariate extension of Elliptic Fourier Analysis (EFA) (See, Kuhl, F. P., and C. R. Giardina. 1982. Elliptic Fourier features of a closed contour. Computer Graphics and Image Processing, 18:236-258; and Ferson, S., F. J. Rohlf, and R. K. Koehn. 1985. Measuring shape variation of two-dimensional outlines. Syst. Zool., 34:59-68), usually used in the analysis of outlines, for use in characterizing motion-driven shape changes is provided. The methods of the present invention represent an advancement of geometric morphometrics at a very fundamental level. They extend the current geometric methods for the analysis of scatters of individual shapes in shape space to allow for the analysis of scatters of sets of shapes in xe2x80x9ctrajectoryxe2x80x9d or motion space.
The multivariate extension of Elliptic Fourier Analysis (EFA) (Kuhl and Giardina, 1982, supra; Ferson et al., 1985, supra), usually used in the analysis of outlines, can address the problem of characterizing motion-driven shape changes. Specifically, the present invention utilizes EFA to parameterize trajectories through shape space resulting from motion. This maps sets of configurations along trajectories to single points in xe2x80x9ctrajectory spacexe2x80x9d that can be used to compare, test, and summarize samples of such trajectories.
A useful approach to solving the problems of the prior art can be found in the use of Elliptic Fourier Analysis (EFA) to model the trajectories of shape changes resulting from repetitive motions. That such an approach is reasonable is illustrated in FIGS. 3A-3C, where the configurations of points connected sequentially throughout single replicates of the hypothetical motion have been joined to form the shape trajectories. FIG. 3A shows all twenty-seven trajectories in the test data discussed above with reference to FIGS. 1A-1C and 2A-2B. For clarity, FIG. 3B shows single trajectories for the model individual of each species of FIGS. 1A-1C and 2A-2B. FIG. 3C illustrates individual variation in shape trajectory for three replicates of the model specimen of brevistrokus of FIG. 1A. Gaps in the trajectories are between the first and last points (unlabeled) collected in the motion cycle. Full periodicity of the shape change, where the shapes of the starting and ending configurations are coincident, is not mandatory, but a useful initial model.
Accordingly, a method for geometrically analyzing motion is provided. The method comprising the steps of: choosing a set of points having at least three individual points to define a single realization of a motion; sequentially collecting Cartesian coordinates of the set of points at different times during the motion from a start point to an end point; treating the collection of sets of points as a sample of the motion; and transforming the sets of points at the different times to a common coordinate system thereby defining a trajectory of the motion.
In a preferred implementation of the method of the present invention, the method further comprises the steps of: choosing a set of points having at least three individual points to define a single realization of a motion; sequentially collecting Cartesian coordinates of the set of points at different times during the motion from a start point to an end point; treating the collection of sets of points as a sample of the motion; transforming the sets of points at the different times to a common coordinate system thereby defining a trajectory of the motion; and calculating elliptic Fourier coefficients describing the trajectory of the motion independent of any difference in the spacing of the different times.